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Theorem ditgex 23335
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
Assertion
Ref Expression
ditgex ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V

Proof of Theorem ditgex
StepHypRef Expression
1 df-ditg 23330 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 itgex 23256 . . 3 ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V
3 negex 10126 . . 3 -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V
42, 3ifex 4101 . 2 if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V
51, 4eqeltri 2679 1 ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1975  Vcvv 3168  ifcif 4031   class class class wbr 4573  (class class class)co 6523  cle 9927  -cneg 10114  (,)cioo 11998  citg 23106  cdit 23329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-nul 4708
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-uni 4363  df-iota 5750  df-fv 5794  df-ov 6526  df-neg 10116  df-sum 14207  df-itg 23111  df-ditg 23330
This theorem is referenced by:  itgsubstlem  23528
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